Translations of "はじめて"

Because here it means "for the first time".
If there is anything that already happen in the past, in this case 買うた completes the whole sentence.
Today, for the first time, I "bought" a Japanese castella cake.
はじめた wouldn't make sense because it would change the sentence to "I started / I began today bought Japanese castella cake....

These are the examples that I would change your answer into another one. Remember I don't know what the real answers.
For the first exercise
２．変わりました
３．もいいです
５．回っている
８．弱いです or 溶けます
14．あった
15．いていません
17．違います
19．ます
21．ではなかった（open）, だった（close）
23．っていました
For the 2nd exercise
３．〇
４．✖（来てませんでした）
５．〇
６．You're right. The model answer is 聴こえました. Used the past tense.
７．〇
９．You're right. The model answer is いました.
10．You're right. The model answer is している, or しています.